Given a cartesian monad on a locally cartesian category one can define a generalized graph and a generalized multicategory of type (-multicategory). We want to impose a “suitability” condition on and which would ensure that one can talk about a free -multicategory. In other words, the forgetful functor
U : (C,T)Multicat \to (C,T)Graph
from -multicategories to -graphs has a left adjoint ; this adjunction induces hence a monad on . Moreover one wants to be able to build new cartesian monads iteratively, namely to ensure the same suitability conditions on the , and so on. This is used for example in some of the possible inductive definitions of opetopes.
A category is suitable (in the sense of Tom Leinster) if
A monad is suitable if
The forgetful functor from the category of -multicategories to the category of -graphs has a left adjoint and the adjunction is monadic; the category of -graphs is then suitable and the induced monad is suitable.
Section 6.5 and appendix D in